Week 9 Reflection - Mathematics & traditional and contemporary practices of making and doing

Sometimes accessing mathematical ways of knowing and doing doesn’t start with the mathematics, it starts with the doing, making, creating and from that can comes an understanding of the mathematics, or a recognition of the mathematics and a desire to know more about the mathematics to deepen understanding of the making…the two way inspiration that Dr. Gerofsky talked about in the introduction.

Many of us grew up with the a combination of the worldview of capitalism - wealth as personal accumulation and as a goal in life, and the value of individuality and self-reliance. This is drastically different from Indigenous worldviews in which wealth is measured in the number of children and grandchildren that you have and in how much you give away. The colonial worldview, that many Canadian colonial children grew up, with is that human beings are on the top of the food chain, and in fact all chains, higher than animals and all things on the land. This is also in drastic contrast to Indigenous worldviews in which the interconnectedness of all things – human and more-than-human, elevate the value of each as a result of the interdependence of all things. In Indigenous worldviews the natural world is not a commodity, it is a relation (Indigenomics by Carol Anne Hilton).

The question, “how does an embodied, mindful AND conceptual way of living balance our lives differently from the mainstream traditions of consumer culture, and the industrial model of schooling that most of us grew up with?” is a big question. When I work with the FPPL and as I try to learn about and bring Indigenous ways of knowing into the classroom, I see connections between embodiment, mindfulness and conceptual ways of knowing. Learning is not only about acquiring knowledge, it is also about remembering. The article I read this week was a short APTN article, ‘The spirit of the medicine will lead us back’: How Avis O’Brien is guiding Elders to weave their first cedar hats. The article talks about how O’Brien used to feel shame about her identity, but when she started working with cedar, things began to change for her. “O’Brien says working with the healing power of cedar, the shame began to lift, and she became open to her cultural practices.” She is passing on her healing by teaching others from her community (particularly elders) to work with cedar as well. 

All cultures have traditional ways of creating and making. This week I was inspired by the Close at Hand video about the only rope walk in use in Norway today. My ancestors are from Norway and so I leaned in a little more to that video, watching the scenery and taking in the process of creating the rope. 

The many viewings and readings this week had me thinking about maker spaces in our schools. These spaces are often filled with lots of bright, shiny, plastic materials. I wonder how we can change the way students view these spaces – or activities, even in our own classrooms. How would our students learning/views change if instead of using collections of things found inside and purchased from a store, we bring natural materials into maker spaces, learning from the land, rather than all the shiny plastic materials. Even better, what if we had the maker spaces outside? If we do use human-made materials, can we present upcycling as a legitimate way of living and using materials, rather than a kind of ‘pretend thing’ we do at school? This has given me new thoughts on maker spaces and new possibilities for how to take math outdoors, using natural, foraged fibres from the land. 

I did the making rope activity this week. I cut down some tall plants and brought them home, only to realize that I had not brought the right kind of plants. They were strawy and brittle and broke as soon as I tried to twist them. So, after walking around unsuccessfully looking for appropriate natural fibres outside, I went to my junk drawer… you know the one that we all have. I was going to use some old ribbons that I had saved, and when I lifted the ribbons out, underneath, I found some straw-type material, I’m not exactly sure what it is, but I used that to make my rope. I followed along with the video, 'The art and geometry of rope making and yarn plying'. At first I started by following the directions as I was seeing them. When Susan said that she was using her right hand, I was using my left. I quickly realized that this was difficult for me, so I undid what I had started. I had to sit facing the same way as Susan was in the video (my screen beside me) to follow what she was doing (initially I tried to translate the movements without changing my position, but I was struggling with transforming the moves). I was also having trouble keeping up with the pace of the video. I slowed the speed of the video down to 50%, then I was able to keep up with the movements and I had a few good chuckles too, as Susan’s slowed down voice sounded… mesmerizing (you should check it out!).

Foraged Fibres

Junk Draw

Rope

As I was making the rope, I was thinking how much my students would love to do this, especially if we started by harvesting fibres from the land and then using them to create a real rope. We could start by looking for the proper type of grass or plant from the land around our school. I know that this activity would create a deeper connection for students with the land. I’m not totally sure about the mathematics we would pull out but for grade one students, we could definitely bring in measurement of our initial plant material (leaves or grass length) and compare it to the length of the final rope. We could consider the pattern of movements in making the rope(i.e. twist, twist, twist, twist, cross over – as Susan says in her video). We could test the weight that the ropes are able to hold (as in one of the ‘how to make a rope videos’), we can look at the curves in the rope, and count the numbers of turns, Doing these activities would broaden students understanding of what mathematics is and how connected mathematics is to everything we do. 

I found the viewings, activities and readings this week inspiring. I do wonder about Sharon Kallis’s ability to take the time to make her own clothing from plants. I admire her using her skills to combat fast fashion but it feels like she is coming from a privileged position to have the time, ability etc. to be able to do this. Although… she did say her nettle shirt was 12 years in the making! That’s perseverance! I was amazed by the video on Weaving the Bridge at Q’eswachaka – the history, practicality, craftsmanship, community connection and cyclical nature of making this bridge were inspiring. It is amazing what people can accomplish when we work together. 

Since this is our last post for this class, the question I have is, "what are one or two things that you have learned or experienced in this course that you will explore more deeply and/or apply in your current practice?"

Week 8 Reflection - Mathematics & fibre arts, fashion arts and culinary arts

This week I attended the last session of the Re-Imagining Mathematics Seminar put on by the E. Lando Virtual Learning Centre (Cynthia was organizing along with Jo-ann Archibald, Joanne Yovanovich and Leyton Schnellert). They discussed Kagann Jad, “The mouse Woman” who is the symbol of the Indigenous Math Network. Mouse woman is like math - helpful and ever present. She always finds a way to help us out. This connected for me with the activities this week. The activities – which were maybe not clearly connected to what we traditionally consider mathematics – had a way of sneaking math in. The math was helpful, if not front and centre.

I had an interesting conversation with a colleague this week regarding how to fit Indigenous “content” into mathematics. My colleague is struggling to see: how to ‘fit it in’, the value of taking time away from learning math skills to ‘play’ with cultural aspects of mathematics, how this all relates to student success – both in math classes at school but also in life after school that requires the pre-requisite math skills/courses etc. I tried (somewhat successfully I think) to bring a different perspective into the conversation. I pushed back on the idea that it is enough to teach Indigenous ‘content’, but suggested we need to rethink the way we teach mathematics, asking how we can bring Indigenous knowledge and pedagogy to the classroom, so that Indigenous (and in fact, all students, ethnomathematically) see themselves reflected in our mathematics classrooms. We also discussed a few ways of doing this that necessitate a shift, not a complete overhaul, of our programs. Saying all of this connected for me with Dr. Gerofsky’s statement in the introduction, “It takes a change of attitude to start noticing the mathematics inherent in all these skills, crafts, and practices, and to connect these physical ways of making with the concept and patterns of mathematical relationships.” Thinking differently about what counts and does not count as mathematics is part of this process of shifting attitudes. Sometimes, like Kagann Jad, the mathematics sneaks in!

The reading I am reflecting on this week is b) Uyen Nguyen (Bridges 2020) Folding fabric: Fashion from origami

The following two quotes from Nguyen, give some insight into the article.

“Fashion loves the juxtaposition of extremes.” (p. 93)

“I wanted to give my artwork functionality.” (p.93)

This paper explores Nguyen’s exploration with folding fabric, origami style, and using rigid paneling and heat settings to maintain the folds. The paper explains the techniques Nguyen uses and the types of designs. 

The techniques used include the following:

• Adhesion of semirigid panels to all regions meant to remain planar and unfolded
• Panels constrained in a way that forces them into the desired configuration
• Adding snap buttons to force some vertices to connect
• Heat setting fabric while in folded configuration
• Sewing vertices together to counteract gravity

Where’s the math?

• Patterns, Geometry – understanding the variable stability of geometries, Fibonacci sequence (Nguyen created a series of Fibonacci skirts), measurement – fabric, weight, temperature, effects of gravity, impact of the distance between folds, dihedral angles, all of the mathematics involved in sewing (I’m sure there are more mathematical connections I have missed)

Here are a few images from the article.

Nguyen ends the article with the following quote. I am including it because in it she states the value of the mathematics behind her design and also expresses the desire for deeper mathematical understanding to improve her designs. It’s a long quote but it sums up the article nicely.

I’ve used fashion and origami to represent various concepts in mathematics. While I use a lot of mathematics in my origami design process, having a more rigorous understanding of the mathematical constraints behind the construction of a fitted garment would enable me to improve my designs moving forwards, both with and without origami elements. I plan to design both clothing that can be fitted and practical as well as garments that are more conceptual and artistic. (Nguyen, 2020, p. 102)

Both this article and the viewing, “How Orbifolds Inform Shibori Dyeing” by Carolyn Yackel talked about the idea of constraints. Learning to work within constraints or using constraints to create something mathematically beautiful. I wonder how often we feel that constraints are negative, rather than embracing how constraints can add to the beauty? 

I experienced working with constraints as I (continuing on with my theme this week of folding) chose to do the activity on Miura Ora Origami (the technique Nguyen uses in her fashion design). 

Here is a bit of the process and the final result:

The first two rows were mind- bending and required resilience. But as I continued to repeat the steps to add more rows, I began to understand what I was doing. Eureka!!! It reminded me how exciting it is to successfully make your way through the struggle! The final two rows I was able to complete without following the steps. I got so much satisfaction in completing this folding activity. I showed everyone my final product. It was only by working within the constraints of the necessary folds that I was able to produce this beautiful product, within the constraints came the freedom of the design. (A binary statement?)

Although this type of origami would be outside the capabilities of young students, there are many simpler types of origami that young students are capable of creating. 

What are the benefits of students learning to creatively work within constraints? 

This week, I’m considering how to bridge the dichotomy between the physical world and  the abstract, conceptual world. Folding paper, spatial reasoning, geometry… when we work with the physical, it strengthens our understanding of the abstract. I’d love to hear your thoughts on this. Many of the activities in this course are doing just that, bringing the physical world and the conceptual world of mathematics together – embodying mathematics! Through movement, drama, film, various kinds of art, poetry, novels the mathematics sneaks in, just like Kagann Jad, helping us out!

Week 7 Reflection - Mathematics & Poetry and Novels

“The dream of a wholly abstract, idealized, disembodied mathematics is simply not achievable; mathematics is a system of human interpretation of the world and has human qualities inextricably woven into its very nature.” (Gerofsky, 2011, p. 14 – quote taken from Radakovic, Jagger & Jao, p. 5)

The article I am reflecting on this week, is Writing and Reading Multiplicity in the Uni-Verse: Engagements with Mathematics through Poetry by Radakovic, Jagger and Jao.

 c) Radakovic, Jagger & Zhao: Writing and reading multiplicity in the uni-verse

The article was full of stops for me. 

Jagger is reporting on an experience in which she was looking for “ways to engage her anxious teacher education students with mathematics” (p. 2). The authors took a poem by Sakaki called, “A Love Letter” (1996) and Jagger invited her ‘math anxious’ students to create a poem, based off the example of Sakaki’s poem. Sakaki’s poem included concentric circles and representation of scale, with an initial increase of the circles by a factor of 10 and the geometric increase extending to the leap to light years. The first two stanzas of the poem are, 

Within a circle of one meter

You sit, pray and sing.

Within a shelter ten meters large

You sleep well, rains sounds like a lullaby.

(It is worth a read, for sure.) In Jagger’s class students were asked to write a poem about their place and connect it to an exploration of place value. The hope was that students would include specific content knowledge. The authors were looking for accurate representation of distance and scale. They were also hoping that students would make real connections between mathematical measurements and their lived experiences.

The authors go on to discuss their interpretation of the student’s poems and resulting considerations of mathematics and poetry. They suggest poetry is a safe way into mathematics. 

I have mixed feelings about this. Poems are highly expressive and personal in a way that other genres of writing are not. Perhaps this is linked with the interpretation (and sometime judgement) of poetry. I think it is why poetry has always been a little scary to me. I feel like I am being more vulnerable when I present my thoughts in poetry form. Another scary aspect for me is that I feel like in order to include mathematics into poetry, my understanding needs to be deep so I don’t make mistakes with the mathematics I include in the poem. On the other hand, I enjoyed creating the Fib poems and the PH4 poem this week and did not feel that pressure – and the experience was only slightly scary as I wondered if I ‘did the poems right’ or if ‘I was missing something’.

What I appreciate about this conversation, is the interdisciplinary nature of mathematical poetry. For the young ones, working an understanding of syllables into an understanding of mathematics, as in the Fib poems, is a unique and authentic way to integrate two content areas. In the introduction, Dr. Gerofsky asks the question, “What ideas do you have about ways to integrate literature with mathematics?” I wonder the same thing. The activities we participated in this week is a way. I’d love to hear others’ ideas of different ways to integrate the two. 

Another stop for me in the article is the author’s discussion regarding the ideas of Barthes (1977). “Barthes proposes that the reading and subsequent interpretation of any text is a writing of a new text. The interpretive possibilities are infinite and depended on the knowledge, experiences and beliefs of the reader” to construct meaning (p. 4). I wonder how this fits in with mathematical content of a poem. How does mathematics in poetry affect the interpretive process? The genre impacts the reading and interpretive process, weaving specific ideas related to the genre into the meaning- making process of the poem. Is the truth seen in the text, or is it open to a multiplicity of meanings based on interpretation? These are questions considered by Trifonas and Jagger (2015). The authors of this article state, “reading of poetry is influenced by the readers’ personal experiences, understandings, and beliefs… we believe the subjective space and the blurring of ‘author’ and ‘reader’ can make poems more authentic, that is meaningful, relatable, and relevant for students” (p. 4).  Perhaps the difference lies in whether we consider the poetry to be ‘mathematical poetry’ or ‘poetic mathematics.’

A question I have is, “can students develop their own conceptual understanding through the exploration of mathematical concepts in the context of poetry?” I think the answer is yes, because I would say that in the process of creating poems this week, my understanding or at least my experiences of Fibonacci deepened. I have also experienced this in my own teaching. I have been experimenting with students using story to show mathematical understanding. By asking students to include mathematics in a story they create, I was able to see conceptual understanding of students; understanding that had been hidden to me when asking students to complete traditional assessment activities. I am aligned with Davis and Renert’s (2014) view of mathematics as described in the article. Mathematics is a “collective, connected, and context dependent enterprise in which the focus is on knowing (something dynamic) rather than knowledge (something static)” (p.6). I resonate with the ideas that mathematics is dynamic…it brings mathematics alive and it becomes playful, rather than static and something to be left on the shelf. When we allow students to play, we are able to access a deeper knowledge of their understanding.

Week 7 Mathematical Poetry Play

I've been playing with the Fib Poetry this week, so I am going to start by posting a few of those. Who knew writing poetry could be so fun?!?

Here is my PH4 Poem

Week 6 Reflection - Mathematics & dance, movement, drama and film

 “Meet mathematics in a different way” (von Renesse, 2019, p.133)

b) Julianna Campbell and Christine Von Renesse (2019) Learning to love math through explorations of maypole patterns

The most provocative and inspiring part of this article for me was the discussion around classroom pedagogy. The inquiry and problem based, constructivist pedagogy in the class, resulted in deep mathematical learning and experiences for the non-math major students taking the course. 

This article is about a mathematics exploration class for first year students with non-math majors. The students explored mathematics within the ribbon colours of maypole dances. The article explores “how seeing maypole dancing through a mathematical lens can excite liberal arts students to deeply explore mathematics” (Campbell & von Renesse, 2019, p. 131). The question the students chose to explore is, “how many non-equivalent ribbon patterns are there, given the number of dancers and the number of colours (of ribbons)?” (p. 132).  

Working together as a class, the students developed shared vocabulary, along with other advancements, allowing them to create representations of the mathematics (ribbon patterns) evident in maypole dancing. In this article, von Renesse and Campbell describe students evolving understanding as they develop these representations and capture mathematical ideas embodied in various iterations of the dance. Rather than inhibiting or limiting their mathematical thinking, the development of representations opened up new possibilities and ways of thinking for the students. An example is evident in this statement, “the students first invented the tree representation since it was the easiest to conceptually make sense of” (p. 136). The article goes on to explain several definitions developed by the students before stating the theorems and proofs the students created about various patterns created through different dances with different coloured ribbons.   

Here is a short 44 second video that shows one of the dances in action. Maypole dance in action

The problems encountered were tackled in community, ideas built upon ideas. For example, language used to explain the activity included, “the first mathematical problem the class encountered…” This shows the social nature and shared engagement with the mathematics, and the acceptance of problems as opportunities to be solved, not work to be overcome. The problems were the work. 

Although not all the examples we have seen in this course of embodied, arts-based, outdoor mathematics are from an Inquiry-based pedagogical stance, the pedagogy and the mathematical opportunities inherent in these embodied activities are complementary, bringing an alternate perspective to the industrial model of the past (hopeful thinking here!). Even the simple idea of mathematics as a social activity, rather than an individual, competitive discipline, is a divergence from the traditional, industrial model. When I think back to my own K-12 mathematics education, I cannot think of even one activity (besides playing games) that was done with others. This paper by Campbell and von Renesse, highlights the value of wrestling out the mathematics in community; relationships with content enhanced by relationships with people (and I would suggest we can expand on this by including relationships with land and the more-than-human world as well).

I felt so lucky this week to connect with the grade one class I had at the beginning of the year before starting my new position in the district. I used the lesson from Malke Rosenfeld’s website, Clap Hands: A Body-Rhythm Pattern Game. As students represented patterns with body movements, I could ‘see’ their brains working to remember and incorporate the movements. Those who understood the pattern, were sometimes searching their memory for the proper movement. Other students, simply participated in the movements, paying little attention to the correct order of the pattern. There was discussion in one of the groups of who should be the leader… navigating the social elements. One boy felt that he should be the “leader” because he knew the pattern and could help people stay on track. One boy yelled excitedly, “it’s like a chain reaction!” Each student was engaged in adding an element to the pattern, when it became their turn. Students were helping each other remember the moves. As a teacher, watching students participate in this activity gave me insight into their understanding of pattern, in a different way than a pencil and paper activity would. It was super valuable, although a bit tricky for me as I swooped in to do the activity and had only a short amount of time. 

At the end of the lesson, I asked students to tell me about their experience with the activity. The answers included, it’s in a circle (I asked groups to form a circle while doing the activity so they could go around the circle), "it started going faster", "it was going a pattern", "it was all going really silly", "you did different actions", "we were making up things that aren’t even real". I then asked the students what this activity had to do with math. Answers included "patterns with people and movements", "each time we added a movement", "there was one movement, then two then three… and math has something to do with numbers and patterns", "you could count how many times the person did it…" 

We did this activity on a Thursday afternoon before a 4 day long weekend, just before students were to go to centres. We could have spent much more time unpacking the activity, but centres was calling and so I dismissed students one at a time to go play and I asked them to give me a word to say about the activity that we did. The words (often students gave me phrases) included: "circle in math", "math comes in numbers", "actually a little fun", "I’d like to do it again", "very, very fun to do the activity", "I really, really like it", "a lot of fun", "we were going in a pattern", "like a circle for math", "everyone in the circle was different", "something was the same- the pattern", "everyone has a different movement", "chain reaction", "like exercise", "the same – we were all in the circle doing different actions", "a circle of people", "it was like going over and over again", "it’s a pattern". (I recorded this debriefing conversation and wish I could upload it for you to hear but I’m having trouble figuring out how to do that.) There was a lot of focus on the idea of the circle, which was interesting and something I would revisit if we were to do the activity again. How would it be different if we did the activity in a line instead of a circle etc? I saw this as a connection to something Dr. Gerofsky wrote in the introduction. “Mathematics deals in abstract forms that are not always easy to realize in the physical world.” These six year olds were working out the abstract movement patterns in the physical world with their bodies. A complex task, indeed, of representing mathematics in the real world.

I was reminded how differently lessons translate when working with actual students, as opposed to trying the lessons out with adults or simply reading about the ideas. I loved doing this activity and saw lots of potential for learning. If I was in the classroom full time, I definitely would have slowed it down, probably done the lesson over a few sessions etc. There was a lot of learning to draw out from this one simple activity, a myriad of potential mathematical concepts to explore.

What I loved is that, as the quote from von Renesse above stated, students were invited to "meet mathematics in a different way".

Here is a short little clip of one student during his turn playing the game. If you could see his eyes, his deep thinking would be even more evident! You can 'see' him thinking through the actions to determine what comes next, as he talks himself through the actions.

The course description (of the class in which the maypole dance exploration occurred), on which the article by Campbell and von Renesse is based, was described as a course "based around meta-goals and doesn't require specific content goals" (p. 132). My questions this week are revolving around these ideas. "How can we teach K-12 with this type of inquiry pedagogy while confidently covering the content of the curriculum?" "How can non-math teachers be supported to integrate content and embodied arts based mathematics, system wide?" "What is an effective way to support teachers on this learning journey?"

Julianna Campbell (one of the authors) was a student in the class. In talking about her mathematics experiences, Julianna said that, "K-12 education failed me" (p.134). She stated that math had "always been a source of struggle, insecurity and general annoyance for me" (p.133). She felt that during this time she was, "faking my way through math, neglecting my natural instincts toward curiosity and inquiry" (p.134). 

What can we do so that students no longer experience this "misperception of math" (p. 134) and instead encounter mathematics in school to be authentic, and a way to foster curiosity? 

Project Outline - Jen Whiffin and Joy Fast

 EDCP 553 Project Outline

Jen Whiffin and Joy Fast

Name of Project - Water Songs, Water Dances: Sound, Movement, and Mathematics 

Grade level - Elementary (K-7), we will house the lessons on our blog Outdoor Math so they can be used by anyone.

Outline:

Guided by the Indigenous Storywork R of Respect, we will be using movement and vocalizations to come to know water better. Mathematics will help us transform these vocalizations and movements into unique songs and dances.

Creating Water Songs Level 1 (Primary): Observing, collecting, representing, classifying, patterning

1. Listen to and engage with water in different contexts (Indoor: water tables, bathtubs and sinks, water fountains. Outdoor: different kinds of rain, puddles, creeks, gutters, oceans…)

2. Collectively create vocalizations that remind them of different sounds.

3. Create words and visuals to help them remember the sounds.

4. Sort and classify sounds (examples: deep, long, soft, fast, slow, light, heavy). Create graphical representation to show classification.

5. Create original sound patterns (water songs) by arranging word visuals. Whole class participates in singing sound patterns.

Creating Water Songs Level 2 (Intermediate): Observing, collecting, representing, measuring, comparing, patterning

1. Same initial start as primary (steps 1-4)

2. “Measuring” sounds by comparing to whole, half, quarter, eighth, sixteenth notes. Choosing sounds to match each sound measurement.

3. Representing various notes by folding/cutting coloured paper. Whole paper = whole note (etc). Each note is represented by a different colour.

4. Students work in partners to compose sound patterns by arranging fractional portions of paper. Glue down to create a bar of their song.

5. Partners grouped with other partners to create longer songs. Groups practice water songs by reading their music, using vocalizations and percussion to punctuate the beginning of each note.

Creating Water Dances (any level): Observing, representing, patterning

1. Observing and collecting water movements in different contexts

2. Representing water movements by analogous body movements

3. Representing body movements with graphics

4. Patterning and arranging graphics to create water dances. Can set to music.

5. Sharing and performing water dance patterns.

We will be using the following “lenses” to guide this project:

Holistic, Multifaceted Approaches to Learning (week 4)

• Related to Ecojustice Mathematics, Indigenous Storywork, FPPL
• The so-called “barrier” between arts/humanities and math/sciences is a modern, western, Eurocentric construct
• Many cultures around the world and over time were enriched by bringing these disciplines together (medieval Europe, Persia, Anishnaabe)

Embodied, multisensory, outdoors and arts-based modalities (week 2 and 5)

• Sight and hearing are not the only ways to sense math
• Embodied metaphors may help students take their learning to greater depths (and heights!)
• “A pedagogy that offers nothing but (a) kind of passive reception of teacher lectures results in an impoverished mathematical education for the majority of students, who do not thrive in such circumstances”
• Use of Cognitive Tools (Egan) and Imaginative Education A brief guide to Imaginative Education

Ecojustice Mathematics: 

• Mathematics has an important role to play in helping everyone construct an understanding of the world.
• Mathematics can and should empower everyone to think for themselves.
• Mathematics enriches the mind, body, and spirit of people by helping them notice and make connections to the living world
• Noticing through mathematics can lead to caring
• Caring and connection can lead to the protection and healing of our world

Indigenous Storywork:

• Education should honour heart, mind, and body connections.
• The 4 R’s of Indigenous Storywork (Reverence, Respect, Responsibility, and Reciprocity) help us better understand our roles as listeners, observers, and teachers. They help us learn well from stories, whether those stories are shared by humans or more-than-humans. 

Bibliography:

Please note that we began this reference list last term. We are deepening our understanding by adding to this reference list. The new references we are adding have been highlighted in yellow.

Adams, K. (2019). Stem education and the miracle of life. In A pedagogy of responsibility: Wendell Berry for Ecojustice Education (pp. 117–135). Routledge, Taylor & Francis Group.

Archibald, J., Fast, J., Fox, S., Nicol, C., Rodger, L., Whiffin, J., & Yovanovich, J. (2021, November 24). Indigenous Storywork and Math Education [Webinar]. In Re-Imagining Mathematics Education Webinar Series. https://elvlc.educ.ubc.ca/initiatives/connecting-math/

Archibald, Jo-ann. (2008). Indigenous Storywork: Educating the Heart, Mind, Body, and Spirit. UBC Press.

Bishop, A. J. 1988 Mathematical Enculturation: A cultural perspective on Mathematics Education, D. Reidel Publishing Company, 100-103. 

https://www.csus.edu/indiv/o/oreyd/acp.htm_files/abishop.htm

Bishop, A. J. (1997). Mathematical Enculturation: A cultural perspective on mathematics education. Kluwer.

Brown, M. W. (2009). The teacher-tool relationship: theorizing the design and use of curriculum materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connections curriculum materials and classroom instruction (PI" 17-36). NewYork, NY: Routltedge.

D’Ignazio, C., & Klein, L. F. (2020). Data feminism. The MIT Press.

Kimmerer, R.W. (2015). Braiding Sweetgrass. Milkweed Editions.

Kuo, M., Barnes, M., & Jordan, C. (2019) Do experiences with nature promote learning? Converging evidence of a cause-and-effect relationship. Frontier in Psychology 10:305. doi: 10.3389/fpsyg.2019.00305

Latremouille, J. (2020). The ecological pedagogy of joy. The SAGE Handbook of Critical Pedagogies (pp. 2-25). 

Macedonia, M. (2019). Embodied learning: Why at school the mind needs the body. Frontiers in Psychology, 10. https://doi.org/10.3389/fpsyg.2019.02098 

Martusewicz, R. A. (2019). A pedagogy of responsibility: Wendell Berry for Ecojustice Education. Routledge, Taylor & Francis Group.

Nicol, C., Dawson, A.J., Archibald, J., & Glanfield, F. (Eds). (2020). Living culturally responsive mathematics with/in Indigenous communities. Brill/Sense Publishers.

Nathan, M. J. (2022). Foundations of embodied learning: A paradigm for education. Routledge.

Wagamese, R. (2019). Medicine walk. McClelland & Stewart.

Wiseman, D., Lunney Borden, L., Beatty, R., Jao, L., & Carter, E. (2020). Whole-some artifacts: (STEM) teaching and learning emerging from and contributing to community. Canadian Journal of Science, Mathematics and Technology Education, 20(2), 264-280.

Wolfmeyer, M., Lupinacci, J., & Chesky, N. (2017). EcoJustice mathematics education: An ecocritical (re)consideration for 21st century curricular challenges. Journal of Curriculum Theorizing, 32(2), 53-71.

Week 5 Part 2 Reflection on the readings and introduction: Integrating embodied arts-based and outdoor mathematics education with more traditional ways of teaching

This week’s theme of integrating arts-based and outdoor mathematics with more traditional ways of teaching has occupied a lot of my thinking this week. I am in a position in my district, where we have found that how we’ve (the district as a whole) been teaching mathematics has not equated with increased mathematical success for students. We need to make changes! The question is what should these changes look like? What do we need to do differently to increase student success in mathematics system wide? These are big questions, that encompass pedagogy, teacher efficacy, curriculum etc. There is not an easy answer and so we are trying different solutions, to see if we will find success for students. There are many bumps along the road. The answer will take time, which is difficult because we want to see student success increase now. Individually, I am drawn to teaching outdoors, to trying new ideas and increasing my capacity as a teacher. This system wide view puts me in a different position than I would be if I was only considering my own classroom, my own setting, my own students and my own strengths and limitations. It is through this lens that I considered the readings this week. 

Teaching mathematics as inquiry often feels like swimming against the tide. I appreciated Dr. Gerofsky’s thoughts regarding what is often considered as traditional ways of teaching mathematics through worksheets and teacher lectures. She said in the introduction, “They (worksheets and lectures) are not bad, they are just not enough.” (emphasis mine). How different would our mathematics classrooms look if we taught math as a “way to think, to come to connect with the world?” (Dr. Gerofsky, introduction). It does not have to be one or the other… but both and…

Although I read all three articles with great interest this week, the article I am primarily reflecting on is 

b) Kelton & Ma: Reconfiguring math settings with whole-body, multi-party collaborations

This article brings together theories of social space, embodied cognition and mathematical activity and development. Kelton and Ma propose the valuable interdependence of setting, embodied activity and mathematical tools and practices as related to meaning making. 

The study upon which the article is based used an instructional design feature called “multi-party, whole-body collaboration.” This means students were using their whole bodies in interaction with one or more people, in situated physical actions involving mathematical content. 

Discussion of Space:

Kelton and Ma discuss the relation between activity and space suggesting that space is not just a backdrop of experience, it is an integral part of the experience. Reading about their thinking on spaces as “complex, historically constituted, dynamically experienced and socially produced settings,” was a stop for me as I connected these ideas with Indigenous perspectives on the value of places and spaces (Kelton & Ma, 2018, p. 178).

Discussion of Embodiment:

The study suggests that we need to move beyond a primary consideration of hands and expand to including other parts of the body, including the whole body. Dr. Gerofsky was cited in the article as an aspiration in this area to, “a more holistic consideration of human bodies” (p. 180). The Sarah Chase video is an example of this in action.

Discussion of Multi-modality:

Kelton and Ma assert, “we are committed to the notion that all mathematical activity is always multi-modal” (p.181). Another stop for me was when the authors share the ideas of Hutchins (2010) who argues that ‘courses of action’ can be ‘trains of thought’. How does this play out in our classrooms? Collaborative mathematical sense-making is pushing beyond words as the main expression of mathematical concepts, to value mathematical thoughts unfolding through action, increasing the visibility of thoughts to others.

Other pertinent ideas discussed by the authors include the importance of design for mathematical activity, which can either “amplify or dampen various bodily multiplicities, possibilities and relations” (p. 182), the individual versus social use of space, and the value of movements both subtle and big.

The aim of this study was to examine how new ways of using the whole body in familiar spaces “can create new opportunities for mathematics learning” (p.182). The study examines two case studies.

Case 1: Walking Scale Number Line (WSNL)

WSNL took place in a gym with lines made on the gym floor using tape. Students became points on the number line and then had to determine the centre and how to navigate their way through changing positions on the line. (Grades 2-8)

Case 2: Whole and Half

This study took place in the student’s classroom with partners. Partner one spread their hands apart and partner two had to place on of their hands at the ? way point between partner one’s hands. This was repeated in different positions around the classroom and with partner one using different configurations. (Grade 5)

The authors determined that reconfiguring classroom space had important consequences for sense-making. Students were given new opportunities for exchanging ideas, boundaries were explored, and imagination activated resulting in “mathematically significant innovations” (p.191).

The study heralds the importance and value of the design of mathematical activities to “leverage inherently embodied and social nature of mathematical thinking and learning” (p.192).

Changing the use of space “appeared to set up different kinds of relationships between mathematical activity and setting…where bodies went determined where the mathematics was” (p.193).

This article, as well as the other articles and introduction from this week, speak to the salience of lesson design. One question I have is, how significantly do the places and spaces in which we DO mathematics factor into our design of mathematical learning experiences. How does WHERE we do mathematics change or influence how we DO mathematics? I believe as Dietiker states, “It necessitates shifting our primary focus from educational outcomes to moving experiences that have potential to compel one toward an end.” (Dietiker, 2015, p. 3). I also ask along with Dietiker, “How might the aesthetic dimension of mathematical stories be understood and improved?” (p. 6) Which leads me to a question I believe we need to ask ourselves as educators when designing lessons, “What is the mathematical story?” 

To circle back to my opening comments on changes needed to increase system wide student success in mathematics, I agree with Riley et al, who discuss the lack of teacher efficacy in implementing connections in mathematics (connections between mathematical concepts, connections between mathematics and other subjects, connections between mathematics and multi-modal experiences). Riley et al. discuss the key benefits of EASY Minds approach to teaching mathematics which incorporates physical activity into mathematics. They say, “key benefits perceived by both students and teachers were increased enjoyment and enthusiasm for mathematics and enhanced opportunities for students’ social, emotional, physical, and cognitive development” (Riley et al., 2017, p.1667). 

The benefits of integrating embodied arts based and outdoor mathematics education with more traditional ways of teaching have been documented. How can we increase teacher efficacy in creating and implementing mathematics programs that encompass these ideals?

Week 5 Part 1 -Activity: Developing mathematics pedagogies that integrate embodied, multisensory, outdoors and arts-based modalities

Initially for the activity this week, I was struggling to find a way to integrate the dance like arm movements of Sarah Chase into an activity appropriate for early primary children. After a conversation with some colleagues, I came up with the following extensions to ‘3 against 2’.

Body Addition

Step 1- Extending the activity 

1. Introduce students to the idea of the position of an arm as representing a number. For students in grade one (my target audience for this activity) working with the numbers 1-3 is the appropriate level of complexity. For example: arms down = 1, arms straight out to the side mid-body level = 2, arms up to the sky = 3. These numbers would stay the same regardless of which arm, so both sides of the body would be the same. 

Practicing – 1. students mirror teacher movements as teacher calls out the number 2. Teacher calls out a number and students show that number with their own arms. 3. Students work with a small group of 3 – one student calls out a number and the other two demonstrate that number with their arms

2. Once students are comfortable with the meaning of the arm placement as representing a number, and have practiced the movements, combine both arms in an addition equation. For example, one arm down and one arm straight up is the addition equation for 1+3 = 4.

3. Students can work with a partner to come up with arm placement for the numbers 4,5,6 and then 2 students together can act as an equation with an equal sign in the middle (perhaps in groups of 3, one person can hold an equal sign written on a small whiteboard or piece of paper). 

4. Students could show their equations to the class and have students figure out the arm position for the numbers 4,5,6 based on the side of the equation they already know.

Step 2 – Sketch of Curriculum Idea

BC Curriculum Connections: There are many connections to BC Curriculum in these activities, below are listed the most salient connections.

• Numbers represent quantity, Number concepts
• Addition and subtraction with numbers to 10 can be modelled concretely, pictorially, and symbolically to develop computational fluency.
• Develop mental math strategies and abilities to make sense of quantities
• Model mathematics in contextualized experiences
• Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving
• Develop and use multiple strategies to engage in problem solving
• Represent mathematical ideas in concrete, pictorial, and symbolic forms

Guiding Questions

1. Can students use their bodies to represent numbers?

2. Will (and if so, how) the embodied experience of using their body to represent numbers, increases students understanding of numbers (i.e. the constancy of numbers) and their ability to work with numbers (adding and subtracting)?

Integrating Embodied Learning

As learners work through the extensions above, the physical movement can be incorporated into more typical classroom mathematics activities in the following ways:

• As students are learning the meaning of the arm position, teacher can demonstrate and students can write the number on a whiteboard and hold it up - this allows teacher to see if students understand the arm position in relation to number, gives students practice printing numerals, adds a written element to the activity
• Another option is for students to use loose parts to represent the number the teacher is demonstrating with their arms (rather than writing it out) I.e. Take the activity outside and students use rocks to show the number the teacher is demonstrating with their arms.
• Similarly, when the activity moves to equations, students can represent the equation in different ways – through loose parts, using a stick and writing it in the dirt (if outside and an appropriate space is available), writing in a math notebook or on a whiteboard.

Possible Extensions

• Challenge students – how can we use our arm positions to show subtraction equations? 
• How can we use our bodies to show bigger numbers?
• Is this a way of subitizing?
• Working with the ideas of more and less – 3 students work together the person in the middle chooses which way to place the ‘more than’ sign based on what their partner is doing.
• Make up a number dance? (Open-ended activity that teacher would use to further the learning, based on what students do…)
• Add a different type of physical activity, outside or in the gym, using different equipment (rather than arms, for instance bouncing a ball) to demonstrate and represent numbers and addition/subtraction

Week 4 Reflections: Mathematics and the arts

Binaries 

Binaries… it is interesting that this week’s introduction started with this exclusionary concept of binaries – one end of the spectrum or the other, as far apart as possible. The idea of binaries as a way in which we see the world was introduced to me in our last class on teaching mathematics for social and ecological justice. Ideas around this have been rolling around in my head since. I notice these ideas around binaries, almost like “stops” that make me pause and consider. Where, why and how did these seemingly competing ideas come from? Last week, while doing the readings, I began making a list of binaries I was coming across. Perhaps I did this to take the ideas off my working memory load, but also so I can remember the many ways our world is divided or considered this way. So, I stopped when the introduction for this week began with a discussion of binaries. It brings me back to perspectives as I was discussing last week. If we see and understand things in a binary way, we miss the interrelated, connectedness of things, ideas, people. We miss opportunities for growth and learning. As Snow worried, “we (will) lack the flexibility of thinking and acting that is needed to solve big world problems” (Introduction, EDCP 553, Week4).

One stop for me in the introduction this week was this pondering by Dr. Gerofsky, “it is worth asking ourselves why contemporary mainstream culture in places like Canada and the US is quick to separate people into ‘math people’ and ‘arts people’, and to look askance at work that bridges these ‘two cultures’." The answer to this question is not simple or easy, it is big and complicated, but I think it is related to the matrix of domination (D'Ignazio & Klein, 2020): the structural domain with laws and policies as well as educational institutions that privilege some paths and people over others, the disciplinary domain that bureaucratically implements and enforces exclusionary binaries, the hegemonic domain in which culture develops and supports ‘oppressive’ ideas, and the interpersonal domain, in which individuals experience narrow pathways to define who they are. 

The Bridges mathematics/arts group is a beacon of light, providing an example of how we can counter divisive binaries with hopefulness, grace and acceptance while maintaining high standards of excellence. One stop for me in the article by Fenyvesi, “Bridges: A world community for mathematical art,” was the following statement about Reza Sarhangi, “Bridges was begun by a many-sided individual” (Fenyvesi, 2016, p. 37). Many sided, multi-modal, multidisciplinary, polyphonic… all ways of being and describing that fight against the binary, including the binary of openness versus academic legitimacy. In speaking about the potential of the Bridges organization, Fenyvesi says, “today, as changes in the world bring about unseen alterations in the structure of knowledge, any form of research, learning, or creativity capable of heightening awareness toward interlocking systems possesses untold value” (p.44), interlocking… not dividing.

a)  Dylan Thomas & Doris Schattschneider (2011) Dylan Thomas: Coast Salish artist, Journal of Mathematics and the Arts, 5:4, 199-211, DOI: 10.1080/17513472.2011.625346

The article I am reflecting on this week is on Dylan Thomas: Coast Salish artist by Dylan Thomas and Doris Schattschneider. Through reading the other articles on Bridges, I came to learn that Schattschneider is an active member of the Bridges community (as is Dr. Gerofsky!). 

Dylan Thomas (Qwul’thilum) is from the Lyackson First Nation from Valdes Island. Thomas did not consider mathematics to be a part of his world beyond completing his high school math courses, until he began to study the work of Escher and recognized the possibilities of tessellation in Coast Salish design. As an apprentice of Rande Cook, Salish art became a focus for Thomas who states, “the mentor-apprentice relationship is very important to all Northwest Coast arts because it is the only way that our traditional arts have survived” (Thomas, 2011, p. 202). Thomas cites Susan Point and her use of geometric symmetry in Salish art as an influence. 

Thomas' art is rich with mathematics, symbolism, history, culture and meaning. One of his works that particularly stood out to me was "Salmon Spirits" which speaks to the depletion of salmon due to climate change and “depicts the overcrowding of salmon in the spirit world” (p. 204). 

Image from http://dylan-thomas.ca/portfolio/prints/

In this article, we hear two voices. Dylan writes about his art and Schattschneider then writes about the mathematics in Dylan’s art. It is an engaging interplay between the two voices and two levels of understanding. Dylan concludes the discussion with this inspiring statement, “Mathematics and especially geometry have been a big part of my art and because of that, a big part of my life. Mathematics has become a huge inspiration to me, especially the way in which nature uses geometry so beautifully. I plan to continue to study the mathematical side of art to see where it takes me in the future” (p. 210). 

How can we take Dylan’s story, the work of others such as Vi Hart (those music videos were amazing!), the Dancing Euclidean proofs from last week and the work of Bridges, to inspire students and give them permission and opportunities to let go off the art/mathematics binary? 

The Bridges Math and Art conference definitely gives us a starting place to be inspired. With my little ones, I was especially interested in the Family Day public workshops, which reminded me of the Virtual Family Math Fair that we contributed to last February. Mahima and I hosted a session for preschooler, titled “Tik Tok Inspired Pattern Dance Party.” 

What I am considering myself, is can we incorporate art and math in a seamless, connected way, rather than as an occasional ‘fun’ activity that we add to our teaching? How can we help students to learn mathematics through art? Should we? How can we bring families onboard so that parents do not perpetuate the ‘math person/ non-math person’ binary with their children? 

For the activity this week, I looked at the Bridges 2013 gallery. The art pieces I choose to interact with are photography pieces by Mohammad Yavari Rad titled “Waves -1” and “Waves -2”.

Image taken from http://gallery.bridgesmathart.org/exhibitions/2013-bridges-conference

I began by investigating and learning more about different types of waves. This was an interesting site for younger students with lots of information about all sorts of different kinds of waves. https://www.mathsisfun.com/physics/waves-introduction.html

There is so much mathematical language and so many mathematical ideas that can be approached through a study of waves including how waves are measured – frequency, wavelength, and amplitude. Waves are found in water, sound, movement of the earth (earthquakes) etc. I learned about the connections between waves, circles, and triangles, and the simplest equation to represent at wave (y-Sin(x). The mathematics of waves is deep!

Throughout the week, I went to a couple of different bodies of water and took some videos/photos of waves. Here are some of my favourites.

After spending too long trying to unsuccessfully upload the video here, I'm going to instead include a YouTube link so you can view the video. 

Rock Drop on Pond: Traverse Waves

Ocean Surface Waves

I love that the art of photography is about different ways that you can see the world, while looking at the same objects… seems like the same can be said for mathematics.

Just for fun... Check out this Public Media Art Wave in Seoul, Korea. Wave in Seoul, Korea. If you scroll down, you can see a 1 minute video of the wave. Not a strong connection to the topic this week but I came across it and thought it was fun to see.

Week 3 Reflections - Math Outdoors

 Changing Perspectives

This week I read all three articles. As I read each article, I kept coming to the idea of perspectives. (This idea has been rattling around in my head since watching the Antonsen talk in week one.) 

The article I am discussing is c) Doolittle, E. (2018). Off the grid. In Gerofsky, S. (Ed.), Geometries of liberation. Palgrave. https://doi.org/10.1007/978-3-319-72523-9_7

The stated purpose of this article is to “apply stress to the grid concept, in order to discover some of its failures, and then propose alternative geometries better suited to our needs in many domains, including education” (Doolittle, 2018, p. 102).

In talking about the grid Doolittle asks, “How we can reflect reality better in our thinking?” (Doolittle, 2018, p. 102). Our reflection of reality comes from the perspective with which we see reality. How does our reflection change when we change our perspective? Are there perspectives that reflect reality in a more authentic way than the “grid” perspective through which we often unconsciously live our lives? Do we see the grid and its structures as being only beneficial, easy, comfortable, familiar, and giving us a sense of control; or do we also consider the possibility of dangers associated with using the grid?

Doolittle proposes failures of the grid, particularly with regards to roads and maps, giving examples of towns/reserves in which he has lived. He brings in Euclidean geometry as an example of a failure of the grid, its value is limited to “small, uniform regions of space” (p.108), essentially ignoring the power of nature. He discusses alternatives to the rectangular grid which include: the hexagon (consider the bees), Penrose tiling, Amman-Beenker tiling, and Riemannian geometry. 

Riemannian geometry gives us another way to consider perspective, taking us outside of the Euclidean geometric idea of grid. Riemannian geometry, Doolittle describes as applying the Copernican Principle that we are not the centre of the universe, to grids. “Accepting all kinds of straight and curved grids as equally valid ways to measure space” (p.111). No grid is better than another, rather we are encouraged to look past the grid to “refocus on the actual underlying geometry of the situation” (Doolittle, p. 111).

Indigenous perspectives are discussed through string figures and the time grids of Indigenous ways of farming. Indigenous perspectives are also brought in as Doolittle describes his approach to the Königsberg Bridges problem. Doolittle asked the question, “what after all this time, haven’t we done that we should have done?” Thus the problem is considered from a different perspective, opening up previously unconsidered possibilities and his answer to this question leads him to a solution.

The following quote, about the grid in relation to agricultural land caused me to stop and consider whether we could apply the same idea to students. “Too often, the specific life, qualities, and character of a particular place become subordinated to the forcefully imposed “evenness” and uniformity of the grid geometry. (Doolittle, p. 104) Are we “forcing” students into grid like conformity that goes against their natural “grain”?

Using a chaotic control analogy, Doolittle brings the discussion back to education. He states, “transformations in the context of education where we have the notion of the teachable moment, in which a few small ‘bursts’ at the appropriate time might accomplish more than months of haranguing… The non-trivial question is how to identify those critical moments, and in which direction to provide the nudge” (p116-117). How does this compare with the control inherent in living within the grid, where we (teachers) are in control, with little space of students?

How does seeing the grid as beneficial or as dangerous change the way we interact with our world?

In describing the path to the source of the river in the Königsberg Bridges problem, Doolittle maintains “the path… would be long and arduous… but it is available to the diligent, the brave, the caring, and the open-minded” (p. 119). It is my hope that as educators, we can be diligent, brave, caring and open-minded.

As I read through the introduction this week, the ideas of roots and connection stood out to me. Maybe it is because I love taking my students into the Delta Watershed and examining the roots of the big trees in the forest. We consider the interconnectedness of the roots, what is happening beneath the soil, the life under the soil, how deep the roots might go in comparison to the height of the tree, the path that the roots take etc… (lots of good mathematics related to roots!). I am drawn to the idea of students understanding the sources and roots of mathematical ideas as well as how this can be explored and discovered by being outside. We explore the fractals that we see everywhere! (Doolittle discussed complexity theory as a superset of fractal geometry.) “What do you notice” and “what do you wonder” questions draw students attention to how mathematics shows itself in the natural world of the forest. How do we move beyond these questions with young students? The land as an entity is a great teacher. 

This week I attempted the activity a few times. My drawings… perhaps I will keep those to myself for now, but… as I was exploring the world through mathematical perspectives, I used my camera to capture images that highlighted for me natural lines and angles – forms that would not fit easily with the grid. The complexity of patterns and lines found in nature are unmatched with human-made things. We only come close when we mimic nature. 

I have not used a lot of whole body movement in teaching mathematics outside, but I am inspired by the Dancing Euclidean proofs to consider ways to allow students to explore their understanding using their bodies.

Week 2 Reflections - Multisensory Mathematics

 As I was reading the introduction for this week, I stopped to mull over the question, “What would mathematics lessons be like if we took multisensory learning seriously?” The principles of UDL came to mind. Ideas about discovering mathematical relationships and using a variety of representations, supported this proposal that multisensory mathematics lessons embrace the principles of UDL (Universal Design for Learning). UDL has 3 guiding principles which include providing multiple means of: 1) Representation – the “WHAT” of learning 2) Expression – the “HOW” of learning and 3) Engagement – the “WHY” of learning.  You can look here for more information. Universal Design for Learning Guidelines

UDL proposes that we start with the belief that all students are capable and aspires to take away stigmas that have separated students with disabilities from the group. Gerofsky wrote in the introduction, “innovations supporting learning for students with sensory impairments will support learning for all” (Gerofsky, 2022). This is UDL in action. How would schools change if we stopped thinking of students as being disabled and instead considered how our environments – social and physical – were disabling. It changes the conversation completely. Can we stop thinking through the binary (false binary) of abled and disabled?

The article I am summarizing this week is c ) Angelika Stylianodou & Elena Nardi (2019), Tactile construction of mathematical meaning: Benefits for visually impaired and sighted pupils. This study’s aim was to “contribute to inclusion and challenge ableism in the mathematics classroom” (Stylianidou & Nardi, 2019, p. 343). The study challenged the effectiveness of using only the typical senses of sight and hearing in the mathematics classroom. Citing the Convention on the Rights of Persons with Disabilities, the paper defined two different types of inclusion: reasonable accommodation and universal design. The study involved a grade 5 classroom with a visually impaired student and drew upon Vygotskian sociocultural theory and the theory of embodied cognition, among others. The study engaged students in exploring shapes through touch first, prior to exploring the shapes through sight. The authors considered the wholistic experience of sight compared to the gradual experience of touch. Results confirmed that both VI and sighted students developed a deeper understanding and more accurate description of the properties of the shape through the sense of touch. A sighted student commented that using touch revealed, “hidden facts on the shapes.” (Stylianodou & Nardi, 2019, p. 348)

Mathematics is probably the subject that teachers struggle the most with implementing the UDL principles, likely due to the Platonic, Cartesian worldview that has been the basis of most mathematics instruction in schools. Multisensory, embodied activities create connections for students, allow students to understand relationships between ideas, and can enable a variety of representations. Using a variety of representation can lead students to see different perspectives and, as Antonsen proposed last week, result in deepening understanding. This may occur through the enhanced metaphors that are develop as students interact in these multimodal, multisensory activities.

These thoughts lead me to consider the difference between knowledge and understanding in mathematics. Has what many would consider to be traditional mathematics instruction, led to knowledge without understanding? 

In multisensory mathematics I see the coming together of theories of learning, use of imaginative education tools (see here for a mathematics example Learning Math through Stories) and practical applications that build rich and engaging opportunities for students to make sense of the mathematics they are learning.

Reflection on Activities:

Although I am acutely aware of “playing with” food, when there are food insecurities all around us, I can see the potential for using the sense of taste (and multiple senses) in mathematics. (And if the food used is actually eaten as it was with these activities, no food insecurity issues.)

One of my boys and I tried the Mathematically Correct Breakfast activity of cutting the bagel together. We were both successful on the first try (although my success was directly related to Emmett helping me!). This activity revealed to me the increase in surface area (and thus a greater cream cheese to bread ratio - Mmmm) in a way that would have been difficult for me to perceive without actually experiencing it. It helped me to "see" this and "taste" it and has me considering it in a way I wouldn't have without experiencing with my senses. 

Making the hexaflexagon was easier for me this time. I had made hexaflexagons for a previous class, worked my way through the productive struggle and found success came easier this time. Eating the hexamex flexagon burrito was just fun and engaging – ripe with possibilities for conversation and thinking mathematically. 

As I considered the Rockets video, I started to brainstorm different kinds of activities with candies. One of my boys is a huge Reese’s Peanut Butter Cup fan. For Christmas, we gave him a box full of all things Reese’s (and there are a lot of Reese's products!). Many activities could be done comparing the different types of cups – the chocolate to peanut butter ratio (it would be interesting to know the difference between the 1/2 pound cup, regular cup and the thin cup), weight to calories etc.

Although we would be unlikely to complete a Reese’s comparison in class, I have used candy to graph and practice addition with Kindergarten and grade one students and they loved it.

These real 3D multisensory objects offer multiple ways for students to experience the mathematics in the world. 

The activities this week have given me lots to think about!